A Complete Proof That No Odd Perfect Numbers Exist, Version v2.0 (Chinese‑English Dual‑Document)

Abstract

This archive contains two separate PDF manuscripts (Chinese and English versions v2.0). This paper presents a complete argument that no odd perfect numbers exist. Based on Euler’s classical structural characterization for odd perfect numbers, we combine multiplicative properties of the divisor‑sum function σ, modular congruence constraints, quadratic‑residue theory, and Zsigmondy’s theorem to progressively narrow down the candidate space for odd perfect numbers. We first prove that the Euler prime p must satisfy p≡1(mod 12). Second, applying Zsigmondy’s theorem we prove that the exponent k belonging to the Euler prime can only be 1. Third, using quadratic‑residue arguments we deduce that p+1 has to be twice a perfect square. Finally, contradictions derived from modulo‑8 and modulo‑12 congruences show there exists no odd integer n satisfying σ(n)=2n. The whole argument relies only on elementary number theory and classical known theorems and does not build upon any unproven assumptions. A Lean‑4 formal‑verification skeleton is provided at the end of this manuscript.